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Principal symmetric ideals were recently introduced by Harada et al. in [The minimal free resolution of a general principal symmetric ideal, preprint (2023), arXiv:2308.03141], where their homological properties are elucidated. They are ideals generated by the orbit of a single polynomial under permutations of variables in a polynomial ring. In this paper, we determine when a product of two principal symmetric ideals is principal symmetric and when the powers of a principal symmetric ideal are again principal symmetric ideals. We characterize the ideals that have the latter property as being generated by polynomials invariant up to a scalar multiple under permutation of variables. Recognizing principal symmetric ideals is an open question for the purpose of which we produce certain obstructions. We also demonstrate that the Hilbert functions of symmetric monomial ideals are not all given by symmetric monomial ideals, in contrast to the non-symmetric case.
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Symmetric group fixed quotients of polynomial rings
Given a representation of a finite group G over some commutative base ring k, the cofixed space is the largest quotient of the representation on which the group acts trivially. If G acts by k-algebra automorphisms, then the cofixed space is a module over the ring of G-invariants. When the order of G is not invertible in the base ring, little is known about this module structure. We study the cofixed space in the case that G is the symmetric group on n letters acting on a polynomial ring by permuting its variables. When k has characteristic 0, the cofixed space is isomorphic to an ideal of the ring of symmetric polynomials in n variables. Localizing k at a prime integer p while letting n vary reveals striking behavior in these ideals. As n grows, the ideals stay stable in a sense, then jump in complexity each time n reaches a multiple of p.
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- Award ID(s):
- 2053288
- PAR ID:
- 10616555
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Journal of Pure and Applied Algebra
- Volume:
- 228
- Issue:
- 4
- ISSN:
- 0022-4049
- Page Range / eLocation ID:
- 107537
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.jpaa.2023.107537
